2. MECHANICS

2.1. Mechanics - the foundation of physics

The first and most important part of the areas of physics is mechanics, which forms a basis for other parts to be presented later. The quantities used here will reappear in many placer - there are many types of forces, but all follow the laws of Newton. Velocity is not only a quantity to be studied for its own sake, but for example the velocity of an electric charge affects how it reacts to magnetic fields. The mechanics studied in this topic is classical mechanics, developed mainly in the time period 1600-1800. A more precise modern theory of mechanics, involving Einstein's theory of relativity and quantum mechanics can be learned later. For most technical applications - including advanced technology like sending a spacecraft to the planet Mars - this classical mechanics is still sufficient.

2.2.Where are we? How far did we go?

We start the physics course with mechanics which deals with questions like where something is, how fast and in what direction it moves, how its motion changes, what causes it, and some consequences of the answers to these questions. All this fits the universal character of physics. An example of this is the quantity speed (described later): a car may drive at a speed, an animal may run or fly at a speed, blood can flow through your veins at a speed, a distant star or galaxy may move towards or away from us at some speed.

Before we get to the quantity speed, we need to describe something more fundamental: where something is. In physics, there are two ways to tell how far something has moved or how far from a certain point it is:

distance = how you went measured along the path you actually took. The tripmeter in a car measures distance. Since the road can be curved it is difficult to say what direction you took, and distance is then a scalar. Common symbol : s

displacement = how far it is from where you started to where you stopped in a straight line. If you look at the map maybe you can find out that the town you drove to is 25 km to the northwest of where you started. This is a vector, which also often has the symbol s.

If only two directions are possible, it is convenient to used positive values for displacements in one direction and negative in the opposite.

Ex. The train moved 500 m forwards and then 200 m backwards.

* If we call the forwards direction positive, the total displacement is 500m + (-200m) = 500 m - 200 m = 300 m.

* If we call the backwards direction positive, we have - 500 m + 200 m = - 300 m.

Note:

·        in both cases we could add the displacements, with their signs. The same formula could have been used: s_total = s₁ + s₂

·        the answers are different although they represent the same motion in the real world. They must be interpreted using the chosen definition of which direction that is positive.

2.3. How fast are we moving?

This can also be described either with or without a direction:

speed = distance/time This is a scalar. Unit ms^-1

velocity = displacement/time This is a vector, same unit.

(note that 1 ms^-1 = 3.6 kmh^-1)

The same formula is used for both (v for velocity or speed, s for distance or displacement, t for time):

v = Ds/Dt [DB p. 4]

The symbol D stands for the change in something = the difference between what it is now and what it was before. In many situations it can be dropped - for example the time for something to happen is the difference between what the clock showed after it and when it started, but if we started a stopwatch from zero when the event started, then the reading on the stopwatch when the event is over equals the time it took. We then often use the formula in the form

v = s / t

If the velocity is constant (both magnitude and direction!), we have what is called uniform motion, UM.

Frames of reference and relative velocity

Example: A boat A moves with 5 ms^-1 downstream, another boat B with 5 ms^-1 upstream in a river flowing 2 ms^-1 relative to the shore. Both move at 5 ms^-1 in the "river's frame of reference", but their speeds in the "shore frame of reference" (or their speeds relative to the shore) are 3 ms^-1 and 7 ms^-1.

2.4. Acceleration

If the velocity changes (magnitude and/or direction), we have an acceleration. We will first focus on cases where something moves along a straight line, but where the speed = the magnitude of the velocity changes. We use these symbols:

u = initial velocity

v = final velocity

t = time to change velocity from u to v

a = acceleration

Dv = v-u = change in velocity

The definition of acceleration is then

a = Dv/Dt [DB p. 4]

where we can write Dv/Dt = (v - u) / t (assuming that t = the time it took for the velocity to change from u to v). Acceleration is a vector and its unit is ms^-2 (which means m/s²). The formula is often written in this form after solving for v:

· a = (v - u) / t multiply both sides with t so

· at = (v - u) = v - u move - u to the left side, letting it change sign

· at + u = v or as below:

v = u + at [DB p. 5]

If the acceleration a is constant, we have uniformly accelerated motion, UAM.

·     Near earth, all things fall down with a gravity acceleration g = 9.81 ms^-2 if we do not think of air resistance.

• For UM we would have a = 0 and v = u + at would become v = u ; the velocity is constant

2.5. Graphs

UM:

·        The graph of velocity as a function of time (velocity on y-axis, time on x-axis) is a horizontal straight line (the velocity is constant). If an object has traveled for the time t with the velocity v, the displacement (how far it as moved) is given by v = s/t => s = vt. This is the of the area (rectangle) under the graph.

·        The graph of displacement as a function of time is a straight line which is steeper the higher the velocity is. Compare this to the graphs of y = x, y = 2x, y = 3x etc where the graph y = kx is steeper the higher k is. Here we have s = vt with s instead of y, v instead of k and t instead of x. The velocity is now the gradient (slope) of the line. This means that you take any two points A and B on the curve and find how much higher B is than A, then divide it by how much further to the left B is than A.

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UAM:

·        The graph of v as a function of t is now a rising straight line starting from u (initial velocity) on the velocity axis. During the time t it reaches the level v (final velocity). The distance traveled is still the area under this graph - now a trapeze (like a triangle on top of a rectangle). This area can be found by adding the areas of the rectangle and the triangle or by finding the mean or average velocity v_m which is (u+v)/2. Since s = v_mt we then get:

s = [(u + v)/2]t

·        The graph of displacement s as a function of time is now not a straight line but a curve bending upwards (getting steeper and steeper - the gradient or slope is still = the velocity, but since this increases all the time, we would need to draw a "help line" (called tangent) and find the gradient = slope of this by choosing two points on it)

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Other types of motion (neither UM or UAM)

If the velocity is not constant (not UM) and the acceleration not constant (not UM) is still true that the travelled displacement is the area under the v-t curve (which may be found with geometry, numerical approximations on a computer, or other methods) and that the velocity at a certain time is the slope of the s-t curve.

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Instantaneous and average values

If one quantity is the gradient (slope) of another (e.g. velocity from displacement or acceleration from velocity) we can graphically find either an average or an instantaneous value. The average value is the change in the vertical coordinate / the change in the horizontal coordinate. The instantaneous value is the "average" value for an infinitely small change in the horizontal coordinate.

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2.6. The 4 equations of uniformly accelerated motion = UAM

We already have

v = u + at [DB p. 5]

and

s = v_mt where v_m = (u+v)/2 [DB p. 5]

We can now

·        replace v in 2) by u +at and get vm = (u+u+at)/2 = (2u + at)/2

·        simplify v_m = (2u +at)/2 = 2u/2 + at/2 = u + ½at

·        to get s = v_mt multiply with t and have t(u + ½at) = ut + ½at²

so we have:

s = ut + ½at² [DB p. 5]

Another possibility is to

·        solve 1) for t which gives t = (v - u)/a

·        replace t in 2) with this, so s =v_mt = v_m(v - u)/a

·        use from 2) that v_m = (u + v)/2 to get s = (u + v)(v - u)/2a

·        let (u +v)(v - u) = (v + u)(v - u) = vv - vu + uv - uu = v² -vu + vu - u² = v² - u²

which all gives us that

·        s = (v² - u²)/2a which we multiply with 2a to get v² - u² = 2as

·        and finally v² = u² + 2as so :

v² = u² + 2as [DB p. 5]

Note that the equations are valid only for constant acceleration!

2.7. Force and mass

Force (a vector quantity)

is the cause of for example

· deformation (stretching, bending, compressing, other). It is measured with a forcemeter (dynamometer, newtonmeter) containing a spiral metal spring which is extended (stretched out) more the greater the force is. Unit : 1 newton = 1 N.

· acceleration = change in velocity per time.

Resultant force (resultant, total force, net force, sum of all forces)

Often several forces act on the same object. If you hold something in your hand, there is a force of gravity pulling it down and a force from your hand upwards which may balance out the downwards force so the resultant is zero. This can be handled by choosing one direction as positive (ex. up) and giving the forces signs accordingly.

Example:

Force of gravity = F_g = - 5.0 N Force of hand = F_h = 5.0 N

Resultant = F_g + F_h = - 5.0 N + 5.0 N = 0

Newton's 3 laws for forces :

Newton I

If the resultant force on an object is zero, its velocity will be constant.

This can mean either of two possibilities:

·        the object is at rest and will remain so as long as the resultant is zero (like the object in your hand).

·        the object has some velocity and will keep it (both direction and magnitude) as long as the resultant is zero. Example : A car comes to a curve where the road is extremelt slippery because of ice. The driver would like to either slow down or change direction, but because of the ice no force can be applied to it horizontally, so it continues out into the forest where forces from trees it collides with slows it down. (This was in the horizontal dimension - in the vertical dimension there is a force of gravity down which is balanced out by the force from the hard ice in the road keeping it from sinking into it).

A free-body diagram = sketch of an object showing the forces acting on it using arrows with a length proportional to the magnitude (if known). Forces (as other vectors) can using trigonometry be resolved into components in two dimensions perpendicular to each other, and the components added separately. The resultant force/ resultant/ total force/ net force can be found using Pythagoras.

Translational equilibrium = a situation where the net force in all dimensions is zero. Example: an object sliding down a slope at constant speed, when the component of the force of gravity down the slope and the force (ex. friction) balance out, and the same is true for the normal force (perpendicular to surface) and the component of the force of gravity perpendicular to the surface (draw diagram, choose labels, resolve into components!).

Newton II

If there is a resultant force F, then there will be a change in velocity = acceleration

which is greater the greater F is, but smaller the greater the mass of the object is.

a = F/m

A larger engine giving a larger net force will increase acceleration

A larger mass will decrease it.

F = ma [DB p. 5]

This means that the unit 1 N = 1 kgms^-2 . Mass is a scalar, but acceleration is a vector, so the force is also a vector.

Newton III

If A acts on B with the force F then B acts back on A with - F

(-F is a force of the same magnitude but opposite direction to F). Examples:

·        A rifle fires a bullet and acts with a force on it accelerating it forwards, but the bullet acts back on the rifle so it recoils

·        A rocket engine in a space ship throws out gases acting with them, and then the gases act back on the rocket with a force forwards (note that the rocket does not "push against the air" to drive it forwards, it works out in empty space).

Mass and weight

·        mass is a property of an object which it has whereever we take it - a 100 kg astronaut is a 100 kg astronaut here or on the moon

· weight is the force of gravity acting on something - on the moon where the force of gravity is weaker, the weight in newtons is lower.

The force of gravity is

F_g = mg

where g = the gravity constant or gravity acceleration = 9.81 ms^-2 on earth, 1.6 ms^-2 on the moon.

· Inertial mass = F/a (where F is resultant force, regardless of what kind of force this - force of gravity, force of hand, force of rocket engine, electrical forces or other).

· Gravitational mass = F/g (near earth) the property of an object which determines how large the force of gravity on the object is.

There is basically no "good" reason why the inertial and gravitational masses should be the same - why the quantity which says how much force of any kind is needed to accelerate an object should be the same as the one which says how strong one particular force (gravity) is. For other the three other fundamental forces (electromagnetic, strong and weak nuclear force) the strength of the force is determined by other quantities (ex. electric charge).

2.8. Friction

Friction

The force of friction is caused by interaction between atoms in the material of a surface and in an object in contact with it. For the force of friction we have

F_fr = m_kN and F_fr < or = m_sN [DB p. 5]

m = positive friction coefficient, without unit, which can be

· kinetic (index k) or dynamic or sliding for moving object (force opposite to velocity)

·        or static (index s) for object at rest (force opposite to net force trying to set it in motion). In this case the value is such that the force of friction balances any net force trying to set the object in motion until some maximum value, when the object "jumps" into motion and the force of friction then is kinetic (with a constant coefficient somewhat smaller than the maximum value of the static one)

N = normal force, the force with which the surface is pressing towards the object (on a horizontal surface N = -F_G so it can be replaced by the force of gravity in a calculation where only magnitudes are involved.

Alternatively: We use different positive-negative directions in the horizontal and vertical dimensions. This means that N or F_N (which is in the vertical dimension, balancing out the force of gravity G or F_G) may be given a different sign when used to calculate the force of friction as the expression mN since m is always positive and the force of friction can be either positive or negative depending our choice of directions. The force of friction is, in principle, not affected by the area of the object which is in contact with the surface.

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For an object on an incline (slope) it must be noted that the normal force is not the opposite of the force of gravity, but of the component of the force of gravity perpendicular to the slope.

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For a moving object, F_fr is in the opposite direction to the velocity. For a static object, it is in the opposite direction to the resultant of all other forces acting on it.

2.9. Work, energy, power

Work and energy

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If the force or a component F_s of it is in the direction of its displacement, the work (a scalar) done is

W = (F_ss =) Fs cosq [DB p. 5]

with the unit 1 joule = 1 J = 1 Nm^-1

The amount of work done is the energy (same unit) converted from one form to another.

In a velocity-time diagram the displacement is the area under the graph since s =vt for UM, for other types of motion the area is not a rectangle but still equal to s. Similarly, in a graph of F_s as a function of s, the area under the graph - rectangle or other - is the work W.

Kinetic energy

·        if a car is accelerated from rest by the constant horizontal force F then the work done is W = Fs = mas; here q = 0

·        from the equation for UAM v² = u² +2as we now get v² = 2as and then a = v²/2s

·        inserting this in W = mas gives W = ½mv² which is "stored" in the moving car, so

E_k = ½mv² [DB p. 5]

Gravitational potential energy

·   if an object falls from the height h the force of gravity does a work W = Fs = mgs = mgh on it:

E_p = mgh [DB p. 5]

These sum of these is the total mechanical energy, which is constant (that is, conserved) unless energy is lost to do work against friction, air resistance or other.

Power

P (= E/t or W/t) = work/time = Fv [DB p. 5]

unit 1 watt = 1 W = 1 Js^-1. Power is the amount of work done or energy transformed from one form to another per time; it can be called the rate of working. "The rate of X" means "how much X per time". Note that for an object moving at a constant speed v the power P = W/t = Fs/t = Fv where F is not the resultant force but the force keeping it in motion despite friction, air resistance etc.

Efficiency

e or h = E_out/E_in or P_out/P_in [not in DB but a similar definition is given in thermal physics, DB p.6]

where E_in is the work or energy supplied and E_out that which is converted to something "useful". What this is depends on the purpose of the device; for a light bulb where a certain amount of electric energy is supplied, the useful energy is that converted to light and the energy converted to heat wasted. For a bread toaster, it is the opposite. Power can be used instead of work or energy since the time t is canceled: P_out/P_in = (E_out/t)/(E_in/t) = E_out/E_in

2.10. Springs

Linear springs

If a spring is extended (pulled out) or compressed (pushed in) a displacement x it acts with a force according to

F = (-) ks [DB p. 5]

A force which follows this type of a formula is called a harmonic force. 

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where k = spring constant, unit Nm^-1 (higher the stronger the spring is);

the minus sign shows that the force of the spring is in the opposite direction to the displacement s from the equilibrium position

Elastic potential energy

When a spring is extended or compressed, work is done on it which can be stored in it as an elastic potential energy. Since the force needed to overcome the force of the spring is not constant but increases linearly the work done = the area under the force graph = ½ * the base * the height = ½ * x * F = ½ * x * kx =

E_elas = ½kx² [DB p. 5]

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2.11. Momentum and impulse

Momentum (linear)

a vector quantity, unit 1 kgms^-1 , is defined as:

p =mv [DB p. 5]

If we define momentum p = mv we can also write NII as F = Dp/t (meaning "net force is the rate of change in the momentum") since initial momentum = mu, final momentum = mv and change in momentum per time = (mv - mu)/t = m(v - u)/t = ma = F. Note that momentum = Fi. 'liikemäärä', Sw. 'rörelsemängd'. Fi. '(voiman) momentti' or 'vääntömomentti' and Sw. '(kraft)moment' or 'vridmoment' all = torque, a quantity to be presented later.

Note: here F is the resultant force

F = Dp/Dt [DB p. 5]

When two objects A and B collide or otherwise interact for the time t and no external force is acting (e.g. the force of friction can is neglected), the total moment is conserved (the same before and after the collision) since

·        N III : A acts on B with F so B acts on A with - F

·        no external forces, so these are the resultant forces on A and B

·        N II for A: - F = ma_A = m(v_A - u_A)/t = (mv_A - mu_A)/t = Dp_A / t

·        N II for B: F = ma_B = m(v_B - u_B)/t = (mv_B - mu_B)/t= Dp_B / t

·        therefore Dp_A/t = - Dp_B/t and Dp_total = Dp_A + Dp_B = 0

·        no change in total momentum means it is the same before and after

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In calculations for problems with two objects colliding, the most useful form of this is

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ [not in DB]

where the formula is adapted according to the situation, e.g. :

·        if object 2 was at rest before the collision then u₂ = 0 and the term m₂u₂ dropped

·        if the objects stay together after the collision, then v₁ = v₂ = v and m₁v₁ + m₂v₂ = (m₁ + m₂)v

·        one direction is chosen positive, and the velocities given positive or negative values accordingly. If a velocity is calculated, the sign shows its direction

Since momentum is a vector we can have collisions in two dimensions where the momentums and/or the velocities are split up into components in two perpenducular dimensions. These are then both conserved m₁u_1X + m₂u_2X = m₁v_1X + m₂v_2X and m₁u_1Y + m₂u_2Y = m₁v_1Y + m₂v_2Y). The components of the momentum are found using trigonometry like for velocities.

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Another useful relation is the following: Since p = mv => p² = m²v² => p²/2m = ½mv² so:

E_k = p² / 2m [DB p. 5]

Impulse

I = FDt = Dp [DB p.5]

(unit 1 kgms^-1 = 1 Ns) where F is the resultant force acting on an object, t the time during which the force acts (can be a very short time for a collision). If the force acting is not constant, the only way to find the impulse and with that the change in momentum is to find the area under the graph of F as a function t. If we find the impulse from the graph, then I =  Dp = m(v-u).

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Elastic collisions

In an elastic collision, e.g. two hard billiard balls colliding and bouncing apart, the total kinetic energy is also conserved.

Example: A billiard ball A with the mass m and velocity u_A collides elastically with another identical ball B at rest. What will happen?

Conservation of momentum: mu_A + mu_B = mv_A + mv_B

=> mu_A = mv_A + mv_B

=> u_A = v_A + v_B

Conservation of kinetic energy: ½mu_A² + ½mu_B² = ½mv_A² + ½mv_B²

=> ½mu_A² = ½mv_A² + ½mv_B²

=> u_A² = v_A² + v_B²

=> (v_A + v_B)² = v_A² + v_B²

=> v_A² + v_B² + 2v_Av_B = v_A² + v_B²

=> 2v_Av_B = 0

which is possible only if v_B or v_A is = 0. The first would require that B is affected by a force without any change in velocity (impossible) so the latter is true.

Inelastic collision

If some kinetic energy is lost, only momentum is conserved (if no external forces act). We must assume that a collision is inelastic unless other information is given. In a completely inelastic collision, all kinetic energy is lost (like two identical cars colliding head with the same speed at forming a wreck at rest together. Since momentum is a vector, the total is conserved - it is zero both before and after!).

2.12. Projectile motion:

Projectile motion = UM horizontally and UAM vertically at the same time

This can be a grenade shot from a cannon, a ball thrown or kicked. The horizontal and vertical motion can be separated - split the initial velocity vector u in such components u_h and u_v:

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We get than u_h = ucosq and u_v = usinq. After that the UM horizontal part and the UAM vertical part are treated with the same equations as before:

Horizontally : u_h = s_h/t = constant = v_h

Vertically : v_v = u_v + a_vt s_v = ((u_v + v_v)/2)t s_v = u_vt + ½a_vt² v_v² = u_v² + 2a_vs_v

where the vertical acceleration a_v = g = 9.81 ms^-2 downwards (given a positive or negative sign depending on whether you chose up or down as positive). The common variable is the time t which can be used to link results from the vertical and horizontal dimensions.

Example: A ball is kicked at the initial velocity u at an angle q on a horizontal field. What is its range?

·        u_v and u_h are obtained as above

·        the time is the same as the time would be for the ball to return to the ground if thrown vertically upwards with u_v

·        s_v = u_vt + ½a_vt² = t(u_v + ½a_vt)=with s_v = 0 and u_v and a_v having opposite signs gives t = 0 or (u_v + ½a_vt) = 0 so t = - 2u_v/a_v (positive)

·        then the horizontal u_h = s_h/t gives s_h = u_ht

At any time during the projectile motion the "final" velocity (the velocity after the object has travelled from the start to the point we are interested in) is found as v = (v_h² + v_v²)^½ and the angle q' to the horizon from v_h/v_v = tan q' giving q' = arctan(v_h/v_v).

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The path followed by an object in projectile motion is part of an upside-down parabola (like the graph to y = -x² ). The reason for this is that the s_v as a function of time is second-degree equation

s_v = u_vt + ½a_vt² or s_v = ½a_vt² + u_vt (compare y = ax² + bx)

and when changing the time values on the horizontal axis to displacement (positition) values with s_h = u_ht for a constant u_h the shape of the graph does not change (compare plotting y = -x² with different scales on the horizontal axis).

2.13. Rotation

Torque

t = Fr sin Q [DB p. 5]

The torque (turning moment, moment of a force) is F times the perpendicular distance r to the pivot (point around which we turn). If F is not at a 90 degree angle to r, we can either take the component of r which is (r sin Q) or the component of F which is (F sin Q). Both give the same formula. Torque is a vector quantity, the possible directions are clockwise and counter-clockwise. When Q = 90^o we use the shorter formula

t = Fr

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We now have two conditions for an object to be at rest:

· translational equilibrium : the resultant force acting on it is zero (in all dimensions, usually no more than two)

· rotational equilibrium : the resultant torque is zero (around all possible pivot points)

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The center of gravity is a point where one can assume that the force of gravity is acting. For objects made of a homogenous material, it is the geometric center (ex. in the middle of a staff). The center of gravity may not be located in the object (e.g. for a ring it is in the center of the ring).

For problem solving:

·        the forces must balance out in all directions (up/down, left/right, others)

·        the torques must balance out around any pivot (choosing ones where the perpendicular distance for a force is zero makes a term disappear!)

Using these principles we try to form a number of equations which give us the values of all unknowns.

2.14. Circular motion

Angles in degrees and radians 

One full turn (revolution) in a circle is 360^o = 2p radians => 1 radian = (180/p)^o. The time or period of a circular motion = T and the speed v = 2pr / T where r is the radius of the circle.

Centripetal and centrifugal force

"Uniform circular motion" = motion at a constant speed v in a circle It is not UM since the direction of the velocity is changing. To keep an object in circular motion we need

·        a centripetal force directed in towards the center of the circle. (ex. whirling a ball in a string: you can not push with a string, only pull)

Because of Newton's III law we then also get

·        a "centrifugal" force acting on what makes the object go in a circle, not on the object itself (ex. an outward force acts on the finger holding the string)

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If we think of an equilateral triangle  with two sides = r and between these a small angle, then third side is ≈ the distance (in a bent curve) traveled by a point on the circle. Since the vector v is always perpendicular to r, it turns the same angle as an imagined string with the length r would have. We note that the 'new' vector v is the first vector v plus the change in velocity Δv which only affects the direction of v, not its magnitude or length. Two equilateral triangles with the same angle between the equal sides are similar to each other in such a way that the ratio between corresponding sides is the same, so for example:

• s/r = Δv/v but since v = s/t we get s = vt and then
• vt/r = Δv/v so multiplying with v and dividing with t we get
• v²/r = Δv/t = a = a_c

which by inserting v = 2pr/T also gives

• a_c = (4p²r²/T²)/r = 4p²r/T²

a = v²/r = 4p²r/T² [DB p. 5]

 which together with F = ma gives the

centripetal force F_c = mv²/r

The centripetal force is not a new fundamental force (like gravity, electromagnetic force, nuclear forces) nor is it a particular force of any more specific type (friction, air resistance, tension in a string - all of which are consequences of mainly electromagnetic forces between atoms and molecules) but rather it is so that different forces (fundamental or their forms in specific cases) act as centripetal force in a certain sitation. Examples:

- for a planet around a sun or a moon or satellite around a planet : gravity

- for a car taking a curve: static fricition between wheels and ground

- for the ball whirled in a string : force of tension in the string

- later : electromagnetic force acting as centripetal force for particles in a magnetic field

[Rotational mechanics, not necessary in the IB

For rotational motion a set of mechanics formulas similar to those for linear mechanics (objects moving in a straight line) can be developed. Instead of the distance or displacement s we can study the angle turned, or the angular displacement q. In radians we have by definition

q = s/r

where r = the radius of the circle and s = the distance covered along its circumference. In a similar way we can define an angular velocity w = q/t (the angle turned per time) and an angular acceleration a = w/t (the change in angular velocity per time).

The results on a rotational motion of a force depend on how far from the center of rotation it is applied, so force will be replaced by torque, t = Fr.

Without proof we will notice that mass also will be replaced by "moment of inertia", I or J where J = mr² if all the mass is at the same distance from the center or axis of rotation. If not, then it can be shown that J follows certain formulas like J = 2/5*mr² for a sphere, J = (1/3)ml² for a bar of length l rotating around one end (like a baseball bat) or J = (1/12)ml² for the bar rotating around its center (like a propeller). Time is the same for linear (translational) and rotational motion. Summary:

TRANSLATIONAL => ROTATIONAL

s => q = s/r

v => w = v/r

a => a = a/r

F => t = Fr

m => J = mr²

Using the "word list" above we can "translate" the known translational formulas into the corresponding rotational ones, for example:

v = u + at => w_final = w_initial + at

s = ut + ½at² => q = w_initialt + ½at²

F = ma => t = Ja

E_k = ½mv² => E_rotational = ½Jw²

p = mv => L = Jw

The rotational or angular momentum L will be relevant in Atomic physics later. We may notice that:

L = Jw = mr²(v/r) = mvr

for an electron in a circular orbit around the nucleus of an atom. ]

2.15. Universal gravitation

Earlier we have always used F_G = mg for the force of gravity. But if we go to another planet or moon g has a different value, and if we move far away from our own planet gravity also gets weaker. A more universal formula (valid everywhere in space) is that the force of gravity between two point masses (small masses) m₁ and m₂ at a distance r from each other is (Newton's law of universal gravitation)

F = Gm₁m₂/r² [DB p. 5]

where G = the universal gravity constant = 6.67 x 10^-11 Nm^-2kg^-2

and the minus sign means that gravity is always attractive

Strictly we should calculate the force of gravity between every possible pair of atom in two larger objects attracting each other, but it can be shown (using 3-dimensional integrals!) that

if the object is a homogenous sphere, for places outside the sphere we get the same result as if we assume that all the mass is in the center of the sphere.

Newton's III. law : if m₁ attracts m₂ with F then m₂ attracts m₁ with an equally big force in the opposite direction.

2.16. Gravitational field and potential

Gravitational field strength (= gravity acceleration)

The gravity acceleration or gravity "constant" g (9.81ms^-2) is not always constant now, but can be calculated for a general case:

Let m₁ be the mass of earth and m₂ that of an object outside earth. F_G = m₂g and the law of universal gravitation give

m₂g = (-)Gm₁m₂/r² => (when m first stands for the mass of a small object and then for that of the planet or other large central body)

g = F/m = Gm/r² [DB p. 4]

This can be called the gravitational field strength and is a vector, towards the center of the earth. Generally for any point in space, where more than one planet contributes,

g = F_resultant/m

Potential energy - the new way

Old way (still OK near earth or near planet with known g-values) : An object falls from rest from the height h₂ down to h₁. With what speed will it reach h₁? The change in potential energy becomes kinetic energy, so

mgh₂ - mgh₁ = ½mv² etc. But this was assuming a constant value for g, which is not correct if it falls from 5000 km to 3000 km above the surface of the earth.

Gravitational potential energy

It can be shown (with integrals) that the gravitational potential energy for an object m₂ at a distance r from a point mass or from the center of a sphere (not the surface!) with the mass m₁ is (same as force but r, not r²):

E_p = -Gm₁m₂/r [DB p. 5]

If we say that the zero level of the potential energy is infinitely far away and the minus sign shows that an object at this distance is bound to the planet m₁ and if put there at rest soon will fall down to it, unless it has or gets energies of other kinds, e.g. kinetic or work done by a rocket engine.

Gravitational potential

To generally describe how much potential energy an object m₂ would have if placed here we can give the potential energy per mass of the object, which is called gravitational potential. It has the unit Jkg^-1 and is defined as V = E_p/m₂ so (when m = mass of planet or large central body):

V = -Gm/r [DB p. 5]

This means that if we know the potential at some point, the E_p which is often useful in calculations is the potential times the mass of the object there. If air resistance can be skipped, it does not make any difference how we move between the levels h₁ and h₂ - the energy we get or which is required is the same (= the force of gravity is a conservative force; total mechanical energy is conserved regardless of how we move. The force of friction is non-conservative.)

Summary

Quantity	At planet surface	In general	Unit
force	F = mg	F = Gm1m2/r^2	N
field intensity	g = F/m	g = Gm/r^2	Nkg^-1 (=ms^-2)
potential energy	Ep = mgh	Ep = -Gm1m2/r	J
potential	V = Ep/m = gh	V = -Gm/r	Jkg^-1

Where only one mass m is indicated in the "in space" versions, it indicates the mass of the planet or other massive central body. The quantity gravitational potential V = gh is rarely used in the "near earth" situation. It could be relevant if the same application could be used near the surfaces of two different planets. E.g. a pump which on earth can pump up water to a height h is more specifically able to move water through a certain gravitational potential difference which leads to different h-values depending on the g-value on the planet in question. This situation can be further complicated by differences in athmospheric pressure on the planets, if the pump mechanism depends on that.]

2.17. Orbital motion

For planets moving around a sun, moons or satellites around a planet, the force of gravity is acting as the centripetal force. We often start calculations by noting that for a satellite with mass m₂ orbiting a planet with mass m₁ at the distance r from the planets center, not its surface we have

F_c = F_G => m₂v²/r = Gm₁m₂/r² => m₁v² = Gm₁m₂/r

For the satellite in a stable orbit we then have:

·        the kinetic energy E_k = ½mv² = Gm₁m₂/2r

·        the potential energy E_p = -Gm₁m₂/r

·        the total mechanical energy E_k + E_p = (Gm₁m₂ - 2Gm₁m₂)/2r = - Gm₁m₂/2r

m17a

Note: the so called free fall or "weightlessness" in a stable orbit does not mean that astronauts do not have any mass in space nor that the force of gravity has been shut off. The force of gravity has not even become very much weaker in an orbit near earth - e.g. 300 km above the planet surface the distance to the center has only increased from maybe 6370 km to 6670 km. The astronauts are "weightless" because the (slightly weaker) force of gravity is acting as a centripetal force, it is needed just to keep them circling the earth in this orbit instead of flying out in space in a straight line. There is no force left over to pull them towards the floor of the spacecraft as the force of gravity does when it stands on the ground.

Escape speed

If a spacecraft is given a high enough speed from the surface of a planet, it may get a positive kinetic energy equal to or higher than the negative potential energy it has when it is "bound" to the planet. It could then move infinitely far away from the planet without ever being pulled back, unless it uses its engine to slow down. The minimum necessary speed for this (disregarding resistance in the planet's atmosphere) can be found using:

E_k + E_p = 0 (for the minimum escape speed, we just about reach infinity with the speed about 0 so the E_k = 0; the E_p is zero at infinity by definition).

so

E_k = - E_p => ½m₁v² = -(-Gm₁m₂/r) => v² = 2Gm₁/r => (when m = mass of planet or large central body)

v_esc = Ö( 2Gm/r) [not in DB]

Note: Since the earth is rotating and spacecraft follows it, it has some kinetic energy before the start. To use this it is favourable to let the rocket start close to the equator (where the ground moves with a higher speed than close to the poles to make a revolution in 24 hours) and towards east, in the direction of rotation. One also wants to have some open sea under the first part of the trajectory (path) so that if the rocket explodes, the pieces do not fall on people. This has led to the choices of location (Florida for the USA, French Guayana for France).

2.18. Kepler's laws

Both Kepler I and II can mathematically be proven as necessary consequences of Newton's law of universal gravity, although this is very advanced.

Kepler I : the planets orbit the sun in ellipses with the sun in one focus

Kepler II : a line from the planet to the sun sweeps the same area in the same time

m18a

Consequence: it must move faster when it passes the focus where the sun is, and is near to it. For earth, this occurs when the axis of the earth is tilted so the southern hemisphere is towards the sun. For this reason the summer is slightly shorter down there and Antarctica colder than Greenland (there is a little more incoming sunlight when the sun is closer in the summer there, but this effect turns out to be less important than the length of the summer).

Kepler III : if a planet (now mass m₂) orbits the sun (now mass m₁) with the time period T at an average distances r to the (center of the) sun, we can by approximating the ellipses to circles get:

T² is proportional to r³ <=> T² µ r³ <=> T² = a constant times r³ or r³ = another constant times T²

Why? The speed = distance/time = 2pr/T so F_c = F_G gives as earlier p. 14

m₂v²/r = Gm₁m₂/r² => m₁v² = Gm₁m₂/r =>

m₂(2pr/T)² = Gm₁m₂/r => (4p²r²/T²) = Gm₁/r =>

1/T² = Gm₁/4p²r³ => T² = 4p²r³/Gm₁ = kr³

which in the data booklet is described as:

T² / R³ = constant [DB p. 5]

Note: Kepler III is valid only for objects rotating around the same central mass, e.g. different planets around a sun or different moons or satellites arounde the same planet. A convenient form of Kepler's III. law for two planets or other objects A and B for which it is valid is:

T_A² / r_A³ = T_B² / R_B³